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Theorem elin2 3372
Description: Membership in a class defined as an intersection. (Contributed by Stefan O'Rear, 29-Mar-2015.)
Hypothesis
Ref Expression
elin2.x  |-  X  =  ( B  i^i  C
)
Assertion
Ref Expression
elin2  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C ) )

Proof of Theorem elin2
StepHypRef Expression
1 elin2.x . . 3  |-  X  =  ( B  i^i  C
)
21eleq2i 2360 . 2  |-  ( A  e.  X  <->  A  e.  ( B  i^i  C ) )
3 elin 3371 . 2  |-  ( A  e.  ( B  i^i  C )  <->  ( A  e.  B  /\  A  e.  C ) )
42, 3bitri 240 1  |-  ( A  e.  X  <->  ( A  e.  B  /\  A  e.  C ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    i^i cin 3164
This theorem is referenced by:  elin3  3373  fnres  5376  funfvima  5769  fnwelem  6246  fz1isolem  11415  isabl  15109  isfld  15537  2idlcpbl  16002  divs1  16003  divsrhm  16005  isidom  16061  lmres  17044  isnvc  18221  ishl  18795  ply1pid  19581  rplogsum  20692  isphg  21411  ishlo  21482  hhsscms  21872  mayete3i  22323  elpredim  24247  elpred  24248  elpredg  24249  sltres  24389  nofulllem5  24431  caures  26579  iscrngo  26725  fldcrng  26732  isdmn  26782  isolat  30024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-in 3172
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